Sign in
12thPass logo
New chatPYQ LibraryDoubtsRank report
Sign in to see Recents

Your guest activity stays on this device

Sign in to save progress →
Sign in

Circle question

2019 · 10 Jan · Shift 2 · Q44
Guest · filters and generic practice availableBrowsing as a guest · PYQ filters and generic practice are available. Sign in only for personalised features and saved progress.
  1. PYQ Library
  2. /JEE Main
  3. /Mathematics
  4. /Circle
  5. /2019 · 10 Jan · Shift 2 · Q44

Circle question

2019 · 10 Jan · Shift 2 · Q44

JEE MainMathematicsCircleMCQ+4 / −1
If the area of an equilateral triangle inscribed in the circle x2 + y2 + 10x + 12y + c = 0 is 27327\sqrt 3273​ sq units then c is equal to :
  1. A
    20
  2. B
    25
  3. C
    −-− 25
  4. D
    13
View written solutionFree

Correct answer: B

  1. Write the circle in standard form

Given circle: x2+y2+10x+12y+c=0x^2+y^2+10x+12y+c=0x2+y2+10x+12y+c=0

Complete the squares: x2+10x=(x+5)2−25x^2+10x=(x+5)^2-25x2+10x=(x+5)2−25 y2+12y=(y+6)2−36y^2+12y=(y+6)^2-36y2+12y=(y+6)2−36

So, x2+y2+10x+12y+c=0x^2+y^2+10x+12y+c=0x2+y2+10x+12y+c=0 becomes (x+5)2−25+(y+6)2−36+c=0(x+5)^2-25+(y+6)^2-36+c=0(x+5)2−25+(y+6)2−36+c=0 (x+5)2+(y+6)2=61−c(x+5)^2+(y+6)^2=61-c(x+5)2+(y+6)2=61−c

Hence the radius is R=61−cR=\sqrt{61-c}R=61−c​

  1. Use the area formula for an equilateral triangle

If the side of the equilateral triangle is aaa, then its area is 34a2=273\frac{\sqrt3}{4}a^2=27\sqrt343​​a2=273​

Cancel 3\sqrt33​: a24=27\frac{a^2}{4}=274a2​=27 a2=108a^2=108a2=108 a=63a=6\sqrt3a=63​

  1. Relate side of inscribed equilateral triangle to circumradius

For an equilateral triangle, circumradius is R=a3R=\frac{a}{\sqrt3}R=3​a​

So, R=633=6R=\frac{6\sqrt3}{\sqrt3}=6R=3​63​​=6

Thus, R2=36R^2=36R2=36

  1. Compare with the circle radius

From the circle, R2=61−cR^2=61-cR2=61−c

Therefore, 61−c=3661-c=3661−c=36 c=25c=25c=25

  1. Check the options

Option A: 202020 ❌
Option B: 252525 ✅
Option C: −25-25−25 ❌
Option D: 131313 ❌

Therefore, the correct answer is B.

PreviousNext

More from Circle

  • The straight line x + 2y = 1 meets the coordinate axes at A and B. A circle is drawn through A, B and the origin. Then the sum of perpendicular distances from A and B on the tangent to the circle at the origin is :2019 · MCQ
  • A square is inscribed in the circle x2 + y2 – 6x + 8y – 103 = 0 with its sides parallel to the coordinate axes. Then the distance of the vertex of this square which is nearest to the origin is :2019 · MCQ
  • If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is 90o, then the length (in cm) of their common chord is :2019 · MCQ
  • A circle touching the x-axis at (3, 0) and making an intercept of length 8 on the y-axis passes through the point :2019 · MCQ
  • If a circle of radius R passes through the origin O and intersects the coordinates axes at A and B, then the locus of the foot of perpendicular from O on AB is :2019 · MCQ
  • If a circle C, whose radius is 3, touches externally the circle, x2+y2+2x−4y−4=0 at the point (2, 2), then the length of the intercept cut by this circle C, on the x-axis is equal to :2018 · MCQ
  • If two parallel chords of a circle, having diameter 4units, lie on the opposite sides of the center and subtend angles cos−1(71​) and sec − 1 (7) at the center respectivey, then the distance between…2017 · MCQ
  • If a point P has co-ordinates (0, − 2) and Q is any point on the circle, x2 + y2 − 5x − y + 5 = 0, then the maximum value of (PQ)2 is :2017 · MCQ